# Related Rates of rocket

## Homework Statement

A rocket is 1/2 miles in the air going 40mi/h a bystander is standing 1 mile away from where the rocket took off straight up. What is the rate of change of the angle that the bystander makes with the rocket.

## The Attempt at a Solution

I set up a triangle and got tan(theta)=y. Then I took the derivative and got dtheta/dt * sec^2(theta) = dy/dt. from here i'm kinda lost it seems like i won't be able to do it now because I have to do this without a calculator so how would I calculate the theta at the time where y=1/2 its not a common angle...?

You're right that it's not a common angle, but do you really need the angle $\theta$? You're looking for $\sec(\theta)$ so why don't you use the Pythagorean Theorem, figure out what your hypotenuse is, and then you can find $\sec(\theta)$ without having to know $$\theta$$

Well, one of your problems is that you do not have enough variables. Try defining the 1/2 miles part x and set up a tan theta= x/1 equation to differentiate.

How do you figure there aren't enough variables? With the derived equation one can very easily solve for $$\frac{d\theta}{dt}$$ which is what we desire. Furthermore, you'll notice that what you suggested, is EXACTLY what physstudent1 did.

I set up a triangle and got tan(theta)=y. Then I took the derivative and got dtheta/dt * sec^2(theta) = dy/dt.

Ok,
define the 1/2 mile part to be x. Now set up the equation tan(theta)=x/1. Differentiate and get sec^2(theta)*dtheta/dt=dx/dt. Solve the triangle at the time when x = 1/2, and you find that the hypotenuse is sqrt(5)/2. Solve for sec(theta)^2. Plug in your 40 mph for dx/dt and rearrange the varibles.

Does that make sense, Kerizhn? I don't know that I did it right, but I get an answer for the problem.

Oh,
Remember to divide your dtheta/dt by 60 to get radians per second change.

Younglearner, I'm not sure if you're trying to show us a different way of doing things here, but you emulated precisely what physstudent 1 did in the original post, only with a different variable name.

Furthermore, you just proved that physstudent1 did indeed have enough variables. I'm not exactly sure what the point of just repeating everything we've already done is...